I find buy down and premium leakage concepts very difficult to explain concisely, but I’ll do my best.
First of all, I think the model solution does actually take into consideration the rate increase, although I would agree that the way the solution is presented is confusing.
For me it feels more natural to calculate the buy down effect first. If the solution would have shown their work better, it would have been more obvious that the $180 comes from the original $150 premium with a 20% increase. The $171.18 is the expected premium with shift assuming the leaner benefit costs 7% less (i.e. 70% shift * 93% * $180 + 30% persist * $180). This is lower than $180 because members are obviously buying down their benefits as given in the problem. The $8.82 difference is the buy down effect, which I think of as the “expected premium without any shift” minus the “expected premium with shift.” The problem in practice is that the “expected premium with shift” doesn’t take into account the difference in claims due to risk – that part is rather unexpected.
I think of premium leakage as the “actual premium with shift” minus the “expected premium with shift” (same as above $171.18). The “actual premium with shift” is not the premium that will be received, but rather the premium desired by taking into account the actual difference in risk profile between shifting & persisting members. This is calculated using the expected claims, i.e. 70% shift * $150 * 93% + 30% persist * $250 = $172.65. There is no 20% increase applied to claims here, because the problem did not say claims were expected to increase – only premiums. The difference is then the premium leakage, $172.65 - $171.18 = $1.47.
Hope that helps. Feel free to question or correct anything I may be misunderstanding.